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In a medium saucepan, bring the milk & chicken broth to a low boil. Add in the cornmeal and lower to a simmer. Stir frequently.
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Heat a large skillet on medium heat. Add the chopped onion and garlic and sauté with a drizzle of olive oil. When the onion is translucent, add in the eggplant. Season with salt and pepper to taste. Let the eggplant cook until softened, about 5 minutes.
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Pour in the tomato sauce and freshly minced parsley. Let the sauce come to a boil, then lower to a simmer and cover. Cook for 10 minutes.
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Finish polenta by adding 4 tbsp butter and salt & pepper to taste. Keep stirring until the consistency is thick, not runny. Once thickened, take off the heat and stir in parmesan.
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To plate, top off the polenta with the tomato sauce, shaved parmesan, and fresh parsley